Question

. The radioactive isotope Uranium-234 decays to Thoriu-230 with a half-life of T. Thorium 230 itself is also radioactive and decays to Radium-226 with a half-life of γ and γ > τ Although Radium-226 is also radioactive its half-life is much longer than T and γ and here we assume that it is relative stable. Consider the scenario when we start with a certain amount of pure Uranium-234, because of this chain of decays, we will get a mixture of the three elements over time, let r(t), y(t), ะ(t) to represent the amount of Uranium, Thorium and Radium at time t (a) Write down the odes describing how x, y, z changes over time and the initial condi- tions. (Hint: you need to first translate half-life into decay rate) (b) Find the solution to the odes and describe how the curve of y(t) looks like (by analyzing its derivative).

Please show your steps clearly.

Answer 1

The decay of radioochive cubstance is governed by the The solution to he given lirst order ode Is x) exp (-Ct) Since, holf life of radicactive Subctance is tg , therefor, 켯 substitute in he expression For χ(t), to obtain, (texp (-C t) 2 In (2 ) ヲ c- ts So, the ode is given a dz dt |In ( 2 )

(o) so, the decay of Urenium is modelled by the ode dt and, the decay OF Thorium is modelled by the ode dt ard, for radium dz In(2) 3) Soluon to ode () is (b) x(t)= χ (0) exp|-(1122 ) The solution to the ode (2) is obtained using Loplace tronsform, i.e In (2) s+ In(2)

z In (2) Tate Laplate transorm of egn. (4) to obhain (In(.xlo) て S n/2 Take inverse taplace transfom, to obtain in(22 , χ (0) in l て In(2) In(2) て

tini kamnunkbry(D)-210-o χ(0) to some initially Now, so, Subshitute, to obtrin z (o) て In(z) t 1-디 itetwise, egn (5) tan alco be simpigied to n/2)) exP

Th plot of ylt) is given as by ean. () it)